Abstract

We study theoretically the spatiotemporal response of a lipid membrane submitted to a local chemical change of its environment, taking into account the time-dependent profile of the reagent concentration due to diffusion in the solution above the membrane. We show that the effect of the evolution of the reagent concentration profile becomes negligible after some time. It then becomes possible to extract interesting properties of the membrane response to the chemical modification. We find that a local density asymmetry between the two monolayers relaxes by spreading diffusively in the whole membrane. This behavior is driven by intermonolayer friction. Moreover, we show how the ratio of the spontaneous curvature change to the equilibrium density change induced by the chemical modification can be extracted from the dynamics of the local membrane deformation. Such information cannot be obtained by analyzing the equilibrium vesicle shapes that exist in different membrane environments in light of the area-difference elasticity model.

During cell life, membranes are submitted to an inhomogeneous and variable environment. Local inhomogeneities can be strongly related to biological processes, which has led to experiments investigating the effect of local modifications on biomimetic membranes Angelova et al. (1999). For instance, in the inner membrane of mitochondria, the enzymes that use the local pH difference across the membrane to synthesize adenosine triphosphate, the cell’s fuel, are located in membrane invaginations called cristae Davies et al. (2011). Experiments on model lipid membranes have shown that a local pH change can induce a local dynamical membrane deformation Khalifat et al. (2008); Fournier et al. (2009); Bitbol et al. (2011); Khalifat et al. (2011); Bitbol et al. (2012), and in particular the formation of cristae-like invaginations Khalifat et al. (2008): membrane shape is tightly coupled to local pH inhomogeneities. Other concentration inhomogeneities in the environment of a cell have a crucial biological role, for instance in chemotaxis or in paracrine signaling. It is therefore of great interest to study the response of a biological membrane to a local modification of its environment.

Motivated by experiments conducted on biomimetic membranes, we have developed a theoretical description of the dynamics of a lipid bilayer membrane submitted to a local concentration increase of a substance that reacts reversibly and instantaneously with the membrane lipid headgroups. We focus on the regime of small deformations, and we treat linear membrane dynamics in the spirit of Ref. Seifert and Langer (1993). While our first works focused on the simple case of a constant modification of the membrane involving only one wavelength Fournier et al. (2009); Bitbol et al. (2011), we have recently extended our theoretical description in order to take into account the spatiotemporal profile of the fraction of chemically modified lipids resulting from the local reagent concentration increase Bitbol et al. (2012). This profile is determined by the diffusion of the reagent in the solution that surrounds the membrane. In Ref. Bitbol et al. (2012), we compared the predictions of this theoretical description to experimental measurements of the deformation of the membrane of giant unilamellar vesicles caused by local microinjection of a basic solution, and we obtained good agreement between theory and experiment.

In the present article, we pursue the theoretical investigation of the effect of a local chemical modification on a lipid membrane. In general, the dynamics that results from a local reagent concentration increase is quite complex, as it involves the evolution of the reagent concentration profile simultaneously as the response of the membrane. This is the case in the experimental data analyzed in Ref. Bitbol et al. (2012), which corresponds to microinjection steps lasting a few seconds. Here, we show that the effect of the evolution of the reagent concentration profile on the membrane dynamics becomes negligible some time after the beginning of the reagent concentration increase, after what the dynamics corresponds to the response of the membrane to a chemical modification imposed instantaneously. We find that studying this regime enables to extract interesting properties of the membrane response.

The article is organized as follows. First, in Sec. 2, we review the linear dynamics of a membrane submitted to a local chemical modification. Then, in Sec. 3, we study separately the dynamics associated with each of the two effects that can arise from a chemical modification, namely a spontaneous curvature change and an equilibrium density change of the external monolayer. We find that a local asymmetric density perturbation between the two monolayers of the membrane relaxes by spreading diffusively in the whole membrane. Intermonolayer friction plays a crucial part in this behavior. Subsequently, in Sec. 4, we treat the general case where both effects are present, and we show how the ratio of the spontaneous curvature change to the equilibrium density change induced by the local chemical modification can be extracted from the dynamics. This ratio cannot be deduced from the study of global modifications of vesicle equilibrium shapes in light of the area-difference elasticity model Lee et al. (1999). Finally, Sec. 5 is a conclusion.

For the article to be self-contained, the present section reviews the linear dynamics of a membrane submitted to a local chemical modification, starting from first principles. The main points of this description were presented in Ref. Bitbol et al. (2012). In that article, we compared theoretical predictions to experimental measurements of the deformation of a membrane submitted to a local and brief pH increase. Here, our aim will be to go further in the analysis of our theoretical description in order to understand the fundamental properties of the response of a membrane to a local chemical modification.

2.1 Monolayer free energy

Our description of the bilayer membrane is based on a local version of the area-difference elasticity membrane model Seifert and Langer (1993); Miao et al. (2002); Bitbol et al. (2011).
We focus on small deformations of an infinite flat membrane, and we denote the upper monolayer by + and the lower one by −.

In the absence of a chemical modification, the local state of each monolayer is described by two variables: the local total curvature c defined on the membrane midlayer, which is common to both monolayers, and the local scaled density r±=(ρ±−ρ0)/ρ0, defined on the midlayer of the membrane, ρ0 being a reference density. The sign convention for the curvature is chosen in such a way that a spherical vesicle has c<0. The free energy f± per unit area in monolayer ± reads Bitbol et al. (2011):

f±=σ02+κ4c2±κc02c+k2(r±±ec)2,

(1)

where σ0 represents the tension of the bilayer and κ its bending modulus, while k is the stretching modulus of a monolayer, and e denotes the distance between the neutral surface Safran (1994) of a monolayer and the midsurface of the bilayer. As we assume that the two monolayers of the membrane are identical before the chemical modification, these constants are the same for both monolayers. The spontaneous curvatures of the two monolayers have the same absolute value c0 and opposite signs, since their lipids are oriented in opposite directions. The expression for f± in Eq. (1) corresponds to a general second-order expansion in the small dimensionless local variables r± and ec, around the reference state which corresponds to a flat membrane with uniform density ρ±=ρ0. It is valid for small deformations around this reference state: r±=O(ϵ) and ec=O(ϵ), where ϵ is a small dimensionless parameter used for bookkeeping purposes, which characterizes the amplitude of the small deformations of the membrane around the reference state. Mathematically, ϵ is considered infinitesimal. Note that in general, both c and r±, which describe local small deformations around the reference state, are functions of time and of position on the membrane.

Let us now focus on the way the membrane free energy is affected by the local chemical modification. We consider that the reagent source, which corresponds to the micropipette tip in an experiment, is localized in the water above the membrane. Besides, membrane permeation and flip-flop are neglected given their long timescales. Hence, the chemical modification only affects the upper monolayer, i.e., monolayer +, and not the lower one. Let us denote by ϕ the local mass fraction of the lipids of the upper monolayer that are chemically modified: ϕ depends on time and position since it arises from the local chemical modification. We assume that the reagent concentration is small enough for ϕ to remain small at every time and position on the membrane, and we characterize this smallness through ϕ=O(ϵ). In order to describe the chemically modified membrane, we have to include the third small variable ϕ in our second-order expansion of f+. We obtain Bitbol et al. (2011):

f+

=

σ02+σ1ϕ+σ22ϕ2+~σ(1+r+)ϕlnϕ+κ4c2

(2)

+κ2(c0+~c0ϕ)c+k2(r++ec)2,

where the constants σ1, σ2, and ~c0 describe the response of the membrane to the chemical modification. These constants depend on the reagent that is injected. Their physical meaning will be explained in the next paragraph. Besides, the non-analytical mixing entropy term ~σ(1+r+)ϕlnϕ has been added to our second-order expansion Bitbol et al. (2011). Note that we assume that the three small dimensionless local variables ϕ, r± and ec are of the same order. In fact, in the present work, the deformation of the membrane and the density variation are caused by the local chemical modification, i.e., they are a response to ϕ, which justifies that ec and r± are of the same order as ϕ. We refer the reader to Ref. Bitbol et al. (2011) for more details on the derivation of Eqs. (1) and (2).

The effect of the chemical modification (i.e., of ϕ) on the upper monolayer is twofold. First, the scaled equilibrium density on the neutral surface of the upper monolayer is changed by the amount σ1ϕ/k to first order. Second, the spontaneous curvature of the upper monolayer is changed by the amount −¯c0ϕ to first order, with ¯c0=~c0+2σ1e/κ. These results are obtained by minimization of the free energy of a homogeneous monolayer with constant mass (see A). Hence, the constants σ1 and ¯c0 describe the linear response of the monolayer equilibrium density and of its spontaneous curvature, respectively, to the chemical modification. This explains the physical meaning of the constants σ1 and ~c0 in Eq. (2). Note that σ2 corresponds to the quadratic response of the membrane to the chemical modification, but it will not have any relevant effect in the following.

2.2 Dynamical equations

The elastic force densities in a monolayer described by the free-energy densities in Eqs. (1–2) have been derived in Ref. Bitbol et al. (2011) to first order in ϵ, using the principle of virtual work. As we focus on small deformations of an infinite flat membrane, it is convenient to describe it in the Monge gauge by the height z=h(r), r∈R2, of its midlayer with respect to the reference plane z=0. Then, ec=e∇2h to second order. Such a description is adapted to practical cases where the distance between the reagent source and the membrane is much smaller than the vesicle radius. The force per unit area of the reference plane, which we call “force density”, then reads to first order in ϵ

p+t

=−k∇(r++e∇2h−σ1kϕ),

(3)

p−t

=−k∇(r−−e∇2h),

(4)

pz

=σ0∇2h−~κ∇4h−ke∇2ra−(κ¯c02−σ1e)∇2ϕ,

(5)

where p±t is the tangential component of the force density in monolayer “±”, while pz=p+z+p−z is the total normal force density in the membrane. In these formulas, we have introduced the antisymmetric scaled density ra=r+−r−, and the constant ~κ=κ+2ke2. These expressions show that both the equilibrium density change and the spontaneous curvature change (i.e., both σ1 and ¯c0) can yield a normal force density, and thus a deformation of the membrane, while only the equilibrium density change can yield a tangential force density and induce tangential lipid flow. An illustration of the role of these force densities in the membrane response to a local chemical modification is provided in Fig. 1.

Figure 1: Qualitative representation of the early stages of the membrane response to an instantaneous chemical modification affecting the upper monolayer (monolayer +). (a): Membrane at equilibrium in the reference state (flat shape, uniform lipid densities in both monolayers). (b): Membrane just after the instantaneous local chemical modification: the membrane has not deformed yet at this stage, and the densities are still uniform. Some lipids of monolayer + are modified: they are represented in dark blue (ϕ represents their local mass fraction; for clarity ϕ is locally large in the figure, but in our work, ϕ=O(ϵ)). Here we assume that the result of the chemical modification is to increase the effective size of their headgroups. Hence, the equilibrium density in monolayer + is locally decreased (σ1<0), while the spontaneous curvature is increased in absolute value (¯c0>0). This results into a normal force density pz>0 at the center of the modified zone of the membrane (see Eq. (5)), and into a tangential force density in monolayer +, oriented toward the exterior of the modified zone (see Eq. (3)). Both are indicated on the figure by arrows. (c) Due to the normal force density, the membrane starts deforming upwards. The maximum height H and the full width at half maximum W of the deformation are indicated. Note that tangential flows also appear due to tangential force densities.

Using the elastic force densities in Eqs. (3–5), it is possible to describe the dynamics of the membrane to first order in the spirit of Ref. Seifert and Langer (1993). Let us outline the derivation of these equations Bitbol et al. (2011, 2012), which is described in more detail in B.
We use a normal force balance for the bilayer (see B.2), which involves pz and the normal viscous stress exerted by the surrounding fluid, which is derived in B.1. Besides, as each monolayer is a two-dimensional fluid, we write down generalized Stokes equations (see B.2), which involve p±t, the two-dimensional viscous stress associated with the lipid flow, the tangential viscous stress exerted by the surrounding fluid (see B.1), and the intermonolayer friction ∓b(v+−v−), where v± is the velocity in monolayer ± and b is the intermonolayer friction coefficient Seifert and Langer (1993); Evans and Yeung (1994). This intermonolayer friction is a tangential force that comes into play when one monolayer slides relatively to the other one2. Finally, we use mass conservation in each monolayer. These dynamical equations are best expressed using two-dimensional Fourier transforms of the various fields involved, denoted with hats: for any field f which depends on r and on time t, ^f is such that

^f(q,t)=∫R2drf(r,t)e−iq⋅r.

(6)

Combining all the above-mentioned equations yields a system of first-order linear differential equations on the two-dimensional variable X=(q^h,^ra):

∂X∂t(q,t)+M(q)X(q,t)=Y(q,t),

(7)

where we have introduced the matrix which describes the dynamical response of the membrane Seifert and Langer (1993); Bitbol et al. (2011):

M(q)=⎛⎜
⎜
⎜
⎜
⎜
⎜⎝σ0q+~κq34η−keq24η−keq3bkq22b⎞⎟
⎟
⎟
⎟
⎟
⎟⎠.

(8)

Here, we have assumed that η2q2≪b and ηq≪b. This is true for all the wavevectors with significant weight in ^ϕ, if the modified lipid mass fraction ϕ has a smooth profile with a characteristic width larger than 1 μm.
Indeed, η=10−3Js/m3 for water, and typically η2=10−9Js/m2 and b=109Js/m4Pott and Méléard (2002); Shkulipa et al. (2006). Besides, the forcing term in Eq. (7) reads Bitbol et al. (2011):

Y(q,t)=⎛⎜
⎜
⎜
⎜
⎜
⎜⎝κ~c0q28η^ϕ(q,t)σ1q22b^ϕ(q,t)⎞⎟
⎟
⎟
⎟
⎟
⎟⎠.

(9)

Eqs. (7-8) show that the membrane deformation is coupled to the antisymmetric density: the symmetry breaking between the monolayers causes the deformation of the membrane. Here, the symmetry breaking is caused by the chemical modification of certain membrane lipids in the external monolayer, i.e., to the presence of ϕ. And indeed, Eq. (9) shows that the forcing term in Eq. (7) is proportional to ^ϕ(q,t).

2.3 Profile of the fraction of chemically modified lipids

Before solving Eq. (7), we need to determine ^ϕ(q,t), which is involved in the forcing term Y(q,t). The profile ϕ(r,t) of the mass fraction of chemically modified lipids in the external monolayer arises from the local reagent concentration increase. We focus on reagents that react reversibly with the membrane lipid headgroups. Besides, we assume that the reaction between the lipids and the reagent is diffusion-controlled (see, e.g., Ref. Eigen (1964)). In other words, the molecular reaction timescales are very small compared to the diffusion timescales. For such a reversible diffusion-controlled chemical reaction, ϕ(r,t) is instantaneously determined by the local reagent concentration on the membrane, which results from the diffusion of the reagent in the fluid above the membrane. Note that these hypotheses are verified in the experiments analyzed in Ref. Bitbol et al. (2012), where the reagent is sodium hydroxide.

We consider that the reagent source is localized in (r,z)=(0,z0>0), which would represent the position of the micropipette tip in an experiment (see Ref. Bitbol et al. (2012)). The cylindrical symmetry of the problem then implies that the fields involved in our description only depend on r=|r|. We focus on the regime of small deformations h(r,t)≪z0, and we work at first order in h(r,t)/z0. Besides, we study the linear regime where ϕ(r,t) is proportional to the reagent concentration on the membrane: denoting by C(r,z,t) the reagent concentration field, we have ϕ(r,t)∝C(r,h(r,t),t). To first order in the membrane deformation, this can be simplified into ϕ(r,t)∝C(r,0,t).

The field C is determined by the diffusion of the reagent from the local source in the fluid above the membrane. In microinjection experiments, there is also a convective transport of the reagent due to the injection, but the Péclet number remains so small that diffusion dominates. Besides, since the membrane is a surface and ϕ≪1, we can neglect the number of reagent molecules that react with the membrane when calculating C. Hence, C can be obtained by solving the diffusion equation

∂tC−D∇2C=S,

(10)

where D is the diffusion coefficient of the reagent in the fluid above the membrane, and the source term reads

S(r,z,t)=S0δ(r)δ(z−z0)θ(t),

(11)

where θ is Heaviside’s function. This corresponds to a constant injection flow from the source starting at t=0. In addition, the membrane imposes a Neumann boundary condition
∂zC(r,h(r,t),t)=0. This relation, which corresponds to a vanishing flux across the membrane, can be simplified to first order into

∂zC(r,0,t)=0.

(12)

The solution to this diffusion problem reads

C(r,z,t)

=∫t0dt′∫R2dr′∫+∞0dz′S(r′,z′,t′)×

abcdefghijklmnopqrG(|r−r′|,z,z′,t−t′)

=S0∫t0dt′G(r,z,z0,t−t′),

(13)

where the causal Green function G of our diffusion problem can be expressed using the method of images Alastuey et al. (2008):

G(r,z,z′,t)=G∞(r,z−z′,t)+G∞(r,z+z′,t),

(14)

where we have introduced the infinite-volume causal Green function of the diffusion equation

G∞(r,z,t)=θ(t)(4πDt)3/2exp(−r2+z24Dt).

(15)

Combining Eqs. (13, 14, 15) provides an analytical expression for C(r,z,t), and for ^C(q,z,t). Since ^ϕ(q,t)∝^C(q,0,t), we thus obtain an analytical expression for ^ϕ, which reads

^ϕ(q,t)∝erf(q√Dt−z02√Dt)cosh(qz0)qz0−sinh(qz0)qz0,

(16)

where erf denotes the error function.

As t→∞, ^ϕ(q,t) converges towards

^ϕs(q)∝e−qz0qz0.

(17)

2.4 Resolution of the dynamical equations

The membrane deformation ^h is given by the solution to Eq. (7) with ^ϕ expressed in Eq. (16) and with the initial condition X(q,t=0)=(0,0), corresponding to a non-perturbed membrane. Hence, we have

q^h(q,t)=∫t0ds[v1e−γ1tA(s)+v2e−γ2tB(s)].

(18)

Here, we have introduced the eigenvalues γ1 and γ2 of the matrix M(q) (see Eq. (8)), the associated eigenvectors V1=(v1,w1) and V2=(v2,w2), and the solutions A and B of the linear system

V1A(t)e−γ1t+V2B(t)e−γ2t=Y(q,t).

(19)

The integral in Eq. (18) and the inverse Fourier transform of ^h can be calculated numerically. This gives the spatiotemporal evolution of the membrane deformation h.

2.5 Dimensionless form of the description

Both for physical understanding and for computational efficiency, it is interesting to put our description in dimensionless form. Using the Buckingham Pi theorem and choosing z0 as the distance unit, bz20/k as the time unit and σ0b2z40/k2 as the mass unit, the ten parameters in our equations yield seven dimensionless numbers:

L1=1z0√κσ0,L2=ez0,Σ0=σ0k,

Δ=Dbk,μ=σ0bz0ηk,Σ1=σ1k,G=¯c0z0.

(20)

The parameters L1 and L2 compare the characteristic lengthscales √κ/σ0 and e of the membrane to z0, while Σ0 is a dimensionless version of the membrane tension σ0. Besides, the parameter Δ quantifies the importance of the reagent diffusion on the membrane dynamics. We will discuss briefly the effect of varying Δ in the following.

In the case of a lipid membrane in water, the only dimensional parameters that can span various orders of magnitudes are σ0 and z0. For σ0≥10−8N/m, i.e., for realistic membrane tensions Evans and Rawicz (1990); Pécréaux et al. (2004), and for z0≥5μm, we will see in the following that the only relevant dimensionless parameters in the dynamics of the membrane are μ and α=−Σ0L21G/(L2Σ1). The first one, μ, then corresponds to the ratio of the two eigenvalues of M(q) for q=1/(2z0) (see Sec. 3.1.2): hence, it is a crucial element of the membrane dynamical response. The second one, α, quantifies the relative weight of the spontaneous curvature change and of the equilibrium density change of the external monolayer due to the chemical modification (see Sec. 4). Note indeed that α∝G/Σ1, and that G and Σ1 describe the two effects of the reagent on the membrane, through ¯c0 and σ1. The effect of varying μ and α will thus be discussed in the following.

For our numerical calculations, we initially take

L1=10−1,L2=10−4,Σ0=10−6,

Δ=21.25,μ=10.

(21)

These values correspond to the injection of NaOH in water (D=2125μm2/sCussler (2009)) from a source at z0=10μm above a floppy membrane with typical constants σ0=10−7N/m (see Ref. Pécréaux et al. (2004) and references therein), κ=10−19J, k=0.1N/m and e=1 nm Safran (1994). The values of the intermonolayer friction coefficient b and of the water viscosity η are those given above. Note that the time unit bz20/k is then equal to one second.

In this Section, we will study the two extreme cases G=0 and Σ1=0. In other words, we will study separately the dynamics associated with an equilibrium density change and with a spontaneous curvature change of the external monolayer. We will see that these two manifestations of the chemical modification of the membrane due to the reagent concentration increase yield different spatiotemporal evolutions of the membrane deformation.
The general case where both effects are present will then be discussed in Sec. 4.

A priori, the dynamics of the membrane deformation is quite complex, as it involves the evolution of the reagent concentration profile, due to diffusion, simultaneously as the response of the membrane.
In order to investigate the effect of the reagent diffusion on the dynamics of the deformation, and to see when the effect of the evolution of the reagent concentration profile becomes negligible, we will compare the two following cases:
(i) the realistic case where ^ϕ(q,t) is given by Eq. (16),
(ii) the theoretical case where the stationary modification ϕs in Eq. (17) is imposed instantaneously: ^ϕ(q,t)=^ϕs(q)θ(t). Note that this case corresponds to the limit of very large Δ.

3.1 Equilibrium density change only

Let us first focus on the case where G=0, in which only the equilibrium density of the upper monolayer is affected by the chemical modification. Fig. 2 shows the evolution with dimensionless time τ=kt/(bz20) of the membrane deformation height H(τ)=h(0,τ) in front of the source, and of the full width at half-maximum W(τ) of the deformation (see Fig. 1(c) for an illustration of the definitions of H and W). The cases (i) and (ii) introduced above are presented. On these graphs, we also show Φ(τ)=ϕ(0,τ)/ϕs(0) and the full width at half-maximum Wϕ of ϕ in case (i).

Fig. 2 shows that the membrane undergoes a transient deformation that relaxes to zero while getting broader and broader. The local asymmetric density perturbation relaxes by spreading in the whole membrane. However, this process is slowed down by intermonolayer friction, so the density asymmetry is transiently solved by a deformation of the membrane Sens (2004); Bitbol et al. (2011). This deformation is downwards if Σ1>0, and upwards if Σ1<0.

Effect of reagent diffusion

In the realistic case (i), at short times, the membrane dynamics is governed by the evolution of ϕ due to the reagent diffusion. Conversely, at long times, once ϕ is close enough to its steady-state profile ϕs, the dynamics of the membrane deformation is similar in case (i) and in case (ii). This can be seen in Fig. 2: first, H(i) and W(i) follow Φ and Wϕ, and then, they have an evolution very similar to those of H(ii) and W(ii). Thus, after some time, the effect of reagent diffusion on the dynamics of the membrane deformation becomes negligible, and the dynamics of the realistic case (i) can be well approximated by that of the simpler case (ii), which corresponds to the response of the membrane to an instantaneously imposed modification.

Figure 2: Dynamics of the membrane deformation in the extreme case G=0, where only the equilibrium density is changed. The values taken for the other dimensionless numbers are those in Eq. (21). Both the realistic case (i) where reagent diffusion is accounted for, and the simpler case (ii) where the chemical modification is imposed instantaneously, are considered. (a) Logarithmic plot of the height of the membrane deformation H and of the fraction Φ of modified lipids in front of the reagent source versus dimensionless time τ. Both in case (i) and in case (ii), H is plotted in units of the extremal value it attains in case (ii). (b) Logarithmic plot of the width W of the membrane deformation and of the width Wϕ of the fraction of modified lipids versus τ. Both W and Wϕ are plotted in units of z0. It can be seen on graphs (a) and (b) that cases (i) and (ii) yield similar dynamics for τ≫τc≈0.5. The thin red (gray) lines correspond to the analytical laws mentioned in the text.

The transition time τc between the diffusion-dominated regime and the membrane-response–dominated regime is determined by the convergence of ϕ to ϕs. As the reagent takes a dimensionless time 1/Δ to diffuse from the source to the membrane, we expect τc∝1/Δ. We studied the dynamics of the deformation height for Δ∈[10−3,103] by integrating Eq. (7) numerically, and we found that this law is very well verified. For our standard value of Δ (see Eq. 21), used in Fig. 2, we have τc≈0.5≈10/Δ.

Analytical insight

Let us discuss analytically the simple case (ii). The long-time behaviors obtained in this case are especially interesting, since they also apply to the realistic case (i). Let us focus on σ0≥10−8N/m, i.e., on realistic membrane tensions, and let us keep the standard values of the other parameters involved in M(q) (see below Eq. (21)), as these parameters cannot vary significantly for a membrane in water. The eigenvalues of M(q), which correspond to the two independent relaxation rates of a perturbation of the membrane with wavelength 2π/q, can then be approximated by

γ1≃kq22b,

(22)

γ2≃σ0q4η,

(23)

for all wavevectors with significant weight in ^ϕs if z0≥5μm. Within this approximation, for q=1/(2z0), we have μ=γ2/γ1.
This leads to a simple expression of the solution of the dynamical equation Eq. (7):

^h(p,τ)∝e−pμτ/4−e−p2τ/2μ−2p^ϕs(p),

(24)

with p=qz0. This expression shows that the only parameter that is relevant in the dynamics is μ. Note that, in the realistic case (i), the value of Δ is relevant too.

In the long-time limit, as the deformation spreads, we have 2p≪μ for all the wavevectors with significant weight in ^h(p,τ). In this case, calculating the inverse Fourier transform of Eq. (24) for r=0 yields

H∼1/√τ,

(25)

and the numerical results for W are in excellent agreement with

W∼√τ,

(26)

as can be seen in Fig. 2(b). This law can also be obtained analytically if ϕs is replaced by a Gaussian.

In the short-time limit, Eq. (24) yields H∼τ and W→2(22/3−1)1/2z0. These asymptotic behaviors, both for the short-time limit and for the much more interesting long-time limit, are plotted in red (gray) lines on Fig. 2.

The transition between the two asymptotic regimes is determined by the value of μ, which is the only parameter that controls the dynamics in case (ii). In particular, it is possible to show that the long-term scaling laws are valid for τ≫max(1,1/μ2). We studied the dynamics of the deformation height for μ∈[10−3,103] by integrating Eq. (7) numerically, and we found that this law is very well verified.

Diffusive spreading of an antisymmetric density perturbation

The long-term scaling W∼√τ shows that the local antisymmetric density perturbation spreads diffusively. Let us emphasize that this diffusive behavior is not related to the diffusion of the reagent in the solution above the membrane, as in the long-term limit, ϕ has reached its steady-state profile ϕs. The fact that the long-term scaling W∼√τ holds in case (ii), where the profile ϕs is established instantaneously, as well as in case (i) (see Fig. 2(b)), illustrates that this scaling law is not related to the reagent diffusion.

The long-term diffusive spreading of the antisymmetric density perturbation can be understood as follows. At long times, the difference between the Stokes equations for each monolayer Eq. (50) and Eq. (51) can be approximated by

k∇(ra−σ1kϕ)+2b(v+−v−)=0,

(27)

in real space. To obtain this equation from the difference between Eq. (50) and Eq. (51), we have used η2q2≪b and ηq≪b (see below Eq. (8)), and also |2eq2^h|≪|^ra−σ1^ϕ/k|, which holds for τ≫1. The last relation can be shown using Eq. (24) in the long-time limit. Using Eq. (27), the antisymmetric convective mass current ja=j+−j− can be expressed to first order as

ja=ρ0(v+−v−)=−ρ0k2b∇(ra−σ1kϕ):

(28)

this current has a diffusive form.
Combining this with the mass conservation relations in Eq. (52) finally yields the diffusion equation

∂tra−k2b∇2(ra−σ1kϕ)=0.

(29)

Hence, the local asymmetry in density between the two monolayers finally relaxes by spreading diffusively, with an effective diffusion coefficient k/(2b). This slow relaxation is due to intermonolayer friction. Note that we can now interpret the dimensionless number Δ defined in Eq. (20) as (half) the ratio of the reagent diffusion coefficient D to this effective diffusion coefficient.

The fact that a local density asymmetry in the two monolayers of a membrane relaxes by spreading diffusively is generic. Here, the asymmetry is due to a local chemical modification of one monolayer, but it can also be caused, e.g., by a sudden flip of some lipids from one monolayer to the other Sens (2004). Our description of this diffusive behavior generalizes that of Ref. Sens (2004), which focused on a perturbation with a spherical cap shape and a uniform density asymmetry.
Note that this diffusive behavior was first mentioned in Ref. Evans and Yeung (1994), within a different theoretical framework, and restricting to axially-symmetric deformations.

3.2 Spontaneous curvature change only

Let us now focus on the second extreme case, where Σ1=0, i.e., where only the spontaneous curvature of the upper monolayer is affected by the chemical modification. Fig. 3 shows that the deformation converges to a deformed profile, in contrast with the previous case: while a local density asymmetry spreads on the whole membrane, a local spontaneous curvature modification leads to a locally curved equilibrium shape. The deformation is upwards if G>0, and downwards if G<0.

Figure 3: Dynamics of the membrane deformation in the extreme case Σ1=0, where only the spontaneous curvature is changed. The values taken for the other dimensionless numbers are those in Eq. (21). Both the realistic case (i) where reagent diffusion is accounted for, and the simpler case (ii) where the chemical modification is imposed instantaneously, are considered. (a) Logarithmic plot of the height of the membrane deformation H and of the fraction Φ of modified lipids in front of the reagent source versus dimensionless time τ. Both in case (i) and in case (ii), H is plotted in units of the extremal value it attains in case (ii). (b) Logarithmic plot of the width W of the membrane deformation and of the width Wϕ of the fraction of modified lipids versus τ. Both W and Wϕ are plotted in units of z0. It can be seen on graphs (a) and (b) that cases (i) and (ii) yield similar dynamics for τ≫τc≈0.5. The thin red (gray) lines correspond to the analytical laws mentioned in the text.

As in Sec. 3.1, in the realistic case (i), the membrane dynamics is governed by the reagent diffusion at short times τ≪τc, while at long times τ≫τc, it is similar in case (i) and in case (ii). The long-time dynamics thus corresponds to the response of the membrane to an instantaneously imposed chemical modification.

In the simple case (ii), the approximations on γ1 and γ2 introduced in Sec. 3.1.2 yield

^h(p,τ)∝(1−e−pμτ/4)^ϕs(p),

(30)

where we have used the notation p=qz0 as above. Calculating the inverse Fourier transform of this function for r=0 yields

H∝μτμτ+4,

(31)

for all τ>0. This analytical law for the deformation height H is plotted in red (gray) lines in Fig. 3(a).

Let us focus on the long-time limit, which is especially interesting, since the results found in case (ii) also apply to the realistic case (i). Eq. (30) shows that in this limit, ^h(p,τ)∝^ϕs(p), so that the long-time profile of the membrane deformation is fully determined by that of ϕ. In particular, for τ→∞, Wϕ→2√3z0 and W→2√3z0.

For τ→0, we find W→2(22/3−1)1/2z0 again. These asymptotic behaviors regarding the deformation width W are plotted in red (gray) lines in Fig. 3(b), both for the short-time limit and for the much more interesting long-time limit.

As in Sec. 3.1, in case (ii), μ is the only relevant parameter in the dynamics of the membrane deformation (see Eq. (30)), and it determines the transition time between the two asymptotic regimes. In the realistic case (i), the value of Δ is relevant too.

Generically, a chemical modification will affect both the equilibrium density and the spontaneous curvature of the upper monolayer. The general solution of Eq. (7) is a linear combination of the two solutions obtained for the two extreme cases: G=0 (see Sec. 3.1) and Σ1=0 (see Sec. 3.2). In this Section, we will only discuss the realistic case (i) where the reagent diffusion is taken into account. As in the two extreme cases, the effect on the membrane dynamics of the evolution of ϕ due to reagent diffusion is crucial for τ≪τc, while it becomes negligible for τ≫τc.

For a lipid membrane in water such that σ0≥10−8N/m and for z0≥5μm, we have seen that in both extreme cases, the dynamics is influenced only by μ and Δ (see Secs. 3.1–3.2). Hence, in the general case, the dynamics is influenced by μ, Δ, and by the parameter

α=−κ¯c0σ1e=−Σ0L21L2GΣ1,

(32)

which quantifies the relative importance of the spontaneous curvature change and of the equilibrium density change of the external monolayer due to the chemical modification. Indeed, Eq. (5) shows that the ratio of the normal force density arising from the spontaneous curvature change to the normal force density arising from the equilibrium density change is equal to −2α. The effect of varying Δ was discussed in Secs. 3.1–3.2, and in addition, Δ cannot vary much for a reagent injected in water above a lipid membrane. Hence, we will focus on the influence of μ and α on the membrane dynamics.

The value of the ratio α of the spontaneous curvature change to the equilibrium density change is a priori unknown, as σ1 and ¯c0, which are involved in Σ1 and G, respectively (see Eq. 20), are unknown. The actual values of σ1 and ¯c0 depend on the reagent as well as on the membrane itself, as these two parameters describe the linear response of the membrane to a reagent (see Eq. 2). Let us assume that α>0, i.e., that the equilibrium density change and the spontaneous curvature change induce deformations in the same direction. This is true, e.g., for a chemical modification that affects the lipid headgroups in such a way that it yields an effective change of the preferred area per headgroup Bitbol et al. (2011). A rough microscopic model, where lipids are modeled as cones that favor close-packing, then yields α≈1, which corresponds to the case where the two effects yield destabilizing normal force densities of similar magnitudes Bitbol et al. (2011). Thus, we expect α≈1.

Fig. 4(a) shows the evolution of H in case (i) for three different values of α. The deformation height features an extremum He, and then a relaxation. This behavior is due to the change of the equilibrium density (see Sec. 3.1). Conversely, the nonzero asymptotic deformation H∞ arises from the change of the spontaneous curvature (see Sec. 3.2). The relative importance of He to H∞ thus depends on α. This means that studying the dynamics of the membrane deformation in response to a local chemical modification provides information on the ratio of the spontaneous curvature change to the equilibrium density change induced by this chemical modification.

In Fig. 4(b)–(c), the ratio He/H∞ is plotted versus α for different values of μ. Indeed, as mentioned above, the membrane dynamics is determined both by α and by μ (at constant Δ). The order of magnitude of He can be estimated assuming an equilibrium density change and/or a spontaneous curvature change of a few percent: He is of order 1−10μm for flaccid membranes, and smaller than 0.1μm if σ0≥10−5N/m. Hence, we choose σ0∈[10−8,10−6]N/m, which yields μ∈[1,100] for z0=10μm: such values are taken in Fig. 4(b)–(c).

Figure 4: (a): Logarithmic plot of the height of the membrane deformation H in front of the reagent source versus dimensionless time τ in the realistic case (i), for different values of the parameter α, which quantifies the relative importance of the spontaneous curvature change to the equilibrium density change. The values taken for the other dimensionless numbers are those in Eq. (21). In each case, H is plotted in units of its asymptotic value in the case α=1. The lines show the asymptotic values H∞ of H for τ→∞ in each case, and He denotes the extremal value of H. (b) and (c): Logarithmic plot of He/H∞ versus α in case (i), for different values of the dimensionless parameter μ, which affects the membrane dynamics as well as α. From down to up: μ=1;μ=100.5≃3.2;μ=10;μ=101.5≃32;μ=102.

Our study shows that one can deduce α, and thus the relative importance of the spontaneous curvature change to the equilibrium density change due to a chemical modification, from the measurement of He/H∞. This is very interesting, given that such information cannot be deduced from the study of static and global membrane modifications Lee et al. (1999); Miao et al. (1994). Indeed, the equilibrium vesicle shapes in the area-difference elasticity model are determined by the combined quantity

¯¯¯¯¯¯¯¯¯¯Δa0=Δa0+2ξcb0,

(33)

which involves both the equilibrium density and the spontaneous curvature. In this expression, Δa0 is the dimensionless preferred area difference between the two monolayers, which is related to the asymmetric equilibrium density change, while cb0 denotes the dimensionless spontaneous curvature of the bilayer, which is related to the asymmetric spontaneous curvature change Miao et al. (1994). The parameter ξ in Eq. (33) is a dimensionless number involving the elastic constants of the membrane. Note that the vesicle shape variations due to global modifications of the vesicle environment are usually interpreted as coming only from a change of the spontaneous curvature, under the assumption that the preferred area per lipid is not modified Lee et al. (1999); Döbereiner et al. (1999); Petrov et al. (1999).

In order to determine the ratio α of the spontaneous curvature change to the equilibrium density change in a practical case, it is necessary to know the value of μ (see Fig. 4(b)–(c)). However, as μ does not involve any parameter that depends on the reagent (see Eq. (20)), it is possible to compare the effects of different reagents on the same membrane, i.e., their values of α, even without knowing the precise value of μ.

We have analyzed theoretically the spatiotemporal response of a membrane submitted to a local heterogeneity, in the realistic case of a local concentration increase of a substance that reacts reversibly and instantaneously with the membrane lipid headgroups. In general, the dynamics of the membrane deformation is quite complex, as it involves the evolution of the reagent concentration profile due to diffusion in the solution above the membrane, simultaneously as the response of the membrane. We have shown that, some time after the beginning of the reagent concentration increase, the effect of the evolution of the reagent concentration becomes negligible.

Studying this regime enables to extract interesting properties of the membrane response. We have shown that a local density asymmetry between the two monolayers relaxes by spreading diffusively in the whole membrane. Intermonolayer friction plays a crucial part in this behavior. In addition, we have shown how the relative importance of the spontaneous curvature change to the equilibrium density change can be extracted from the dynamics of the membrane response to the local chemical modification. This is a significant result since such information cannot be deduced from the study of a static and global modification using the area-difference elasticity model.

Our description provides a theoretical framework for experiments involving the microinjection of a reagent close to biomimetic membranes. In Ref. Bitbol et al. (2012), we used the theoretical model presented here to analyze experimental results corresponding to brief microinjections of a basic solution in the regime of small deformations, and we obtained good agreement between theory and experiment.
The results of the present theoretical work show that it would be interesting to conduct experiments with a continuous injection phase, since the ratio α of the spontaneous curvature change to the equilibrium density change could then be determined for various membrane compositions and reagents.

Acknowledgements

We thank Miglena I. Angelova for motivating us to study this subject. We thank Miglena I. Angelova and Nicolas Puff for interesting discussions.

Appendix A Spontaneous curvature and equilibrium density

The spontaneous curvature and the equilibrium density in monolayer + can be obtained by minimizing f+/ρ+ with respect to r+ and c for a homogeneous monolayer with constant mass. First, using the expression of f+ in Eq. (2), the minimization with respect to r+ gives, to first order in ϵ:

r+eq=σ02k+σ1ϕk−ec.

(34)

Then, the minimization with respect to c yields to first order, using Eq. (34):

ceq=−c0−¯c0ϕ−σ0eκ,

(35)

with ¯c0=~c0+2σ1e/κ (as defined in the main text). Note that, since we assume that r+=O(ϵ) and ec=O(ϵ), we must have c0e=O(ϵ) and σ0/k=O(ϵ) for our description to be valid for the values of r+ and c that minimize f+/ρ+. This property has been used to simplify the results of the minimization.

The scaled lipid density r+n on the neutral surface of the monolayer is related to r+ through r+n=r++ec to first order. This relation arises from the geometry of parallel surfaces Safran (1994), given that the membrane midlayer and the monolayer neutral surface are parallel surfaces separated by a distance e. Hence, Eq. (34) can be rewritten as

r+n,eq=σ02k+σ1ϕk.

(36)

This result is independent of the curvature c, contrary to that in Eq. (34). Indeed, by definition, on the neutral surface, curvature and density are decoupled Safran (1994), while these two variables are coupled on other surfaces.

Eq. (36) shows that, due to the chemical modification, the scaled equilibrium density on the neutral surface of monolayer + is changed by the amount

δr+n,eq=r+n,eq(ϕ)−r+n,eq(0)=σ1ϕk

(37)

to first order. Besides, Eq. (35) indicates that the spontaneous curvature of monolayer + is changed by the amount

δceq=ceq(ϕ)−ceq(0)=−¯c0ϕ

(38)

to first order.

One might wonder why the spontaneous curvature ceq found by minimization is not simply −c0 for ϕ=0 (see Eq. (35)). This is due to the fact that we work on the membrane midsurface, which is more convenient to study the membrane dynamics. If the monolayer free energy had been originally written using variables and constants defined on the neutral surface, then the spontaneous curvature found by minimization would correspond exactly to the constant cn0 which plays the part of c0 when everything is defined on the neutral surface.

Appendix B Derivation of the dynamical equations

b.1 Hydrodynamics of the fluid above and below the membrane

We wish to determine the velocity field W±=(w±,W±z) in the fluid above (+) and below (−) the membrane. This flow is caused by the deformation of the membrane and by the lateral flow in the membrane: mathematically, it is determined by the boundary conditions corresponding to the continuity of velocity at the interface between the fluid and the membrane.

Given the short lengthscales considered, the dynamics of W±(r,z,t) can be described using Stokes’ equation. Adding the incompressibility condition, we have:

η(−q2+∂2z)^W±=(iq+ez∂z)^P±,

(39)

iq⋅^w±+∂z^W±z=0,

(40)

where P± is the excess pressure field in the fluid: the total pressure is P±=P0+P±, where P0 is a constant and P± goes to zero far from the membrane. As in the main text, hats indicate two-dimensional Fourier transforms. Taking the scalar product of q and Eq. (39), and using Eq. (40) yields

^P±=−η∂z^W±z+ηq2∂3z^W±z.

(41)

Taking the scalar product of ez and Eq. (39), and using Eq. (41) yields

q4^W±z−2q2∂2z^W±z+∂4z^W±z=0.

(42)

Solving these equations, with the boundary conditions at infinity: ^W±z→0 for z→±∞, and at the water–membrane interface: ^W±z(q,0,t)=∂t^h(q,t) and ^w±(q,0,t)=^v±(q,t), finally yields

^P±

=±2η[q(∂t^h)∓iq⋅^v±]e∓qz,

(43)

^w±

={^v±−[iq(∂t^h)±q^v±]z}e∓qz,

(44)

^W±z

={∂t^h±[q(∂t^h)∓iq⋅^v±]z}e∓qz.

(45)

For our study of the dynamics of the membrane, what is needed is the stress exerted by the fluid on the membrane, i.e., since we are working at first order in the membrane deformation, the stress in z=0. The viscous stress tensor of the fluid is defined by

Tαβ=−(P±+P0)δαβ+η(∂βW±α+∂αW±β),

(46)

where α∈{x,y,z} and β∈{x,y,z}. Hence, using Eqs. (43, 44, 45), we obtain

^T+zz(z=0)−^T−zz(z=0)

=−4ηq∂t^h,

(47)

^T±tz(z=0)

=∓2ηq^v±,

(48)

where we have introduced the tangential part T±tz=(T±xz,T±yz) of the stress tensor of the fluid above and below the membrane.

b.2 Membrane linear dynamics

The first dynamical equation we use is a balance of forces per unit area acting normally to the membrane. It involves the normal elastic force density in the membrane given by Eq. (5) and the normal viscous stresses exerted by the fluid above and below the membrane, which are derived in B.1 (see Eq. (47)). It reads:

−(σ0q2+~κq4)^h+keq2^ra+κ~c02q2^ϕ−4ηq∂t^h=0,

(49)

where η denotes the viscosity of the fluid above and below the membrane.

Besides, as each monolayer is a two-dimensional fluid, we write down generalized Stokes equations describing the tangential force balance in each monolayer. The first force involved is the density of elastic forces given by Eqs. (3) and (4). The second one arises from the viscous stress in the two-dimensional flow of lipids. The third one comes from the viscous stress exerted by the water, which is derived in B.1 (see Eq. (48)). The last force that has to be included is the intermonolayer friction Evans and Yeung (1994). We thus obtain:

−ikq(^r+−eq2^h−σ1k^ϕ)

−(η2q2+2ηq)^v+

−b(^v+−^v−)=0,

(50)

−ikq(^r−+eq2^h)

−(η2q2+2ηq)^v−

+b(^v+−^v−)=0,

(51)

where v± denotes the velocity in monolayer ±, while η2 is the two-dimensional viscosity of the lipids and b is the intermonolayer friction coefficient Evans and Yeung (1994).

Finally, we use the conservation of mass in each monolayer to first order: